Let us now return to Fig. 30. Recall, that as the quality-factor
is gradually increased, the time-asymptotic orbit of the
pendulum through phase-space undergoes a sudden transition, at , from a
left-right symmetric, period-1 orbit to a left-favouring, period-1 orbit. What happens if
we continue to increase ? Figure 39 is basically a continuation of Fig. 30.
It can be seen that as is increased the left-favouring, period-1 orbit gradually
evolves until a critical value of , which is about , is reached.
When exceeds this critical value the nature of the orbit undergoes another sudden change:
this time
from a left-favouring, period-1 orbit to a left-favouring, period-2 orbit.
Obviously, the
change is
indicated by the forking of the curve in Fig. 39. This type of transition
is termed a period-doubling bifurcation, since it involves a
sudden doubling of the
repetition period of the pendulum's time-asymptotic motion.

We can represent period-1 motion schematically as , where represents a pattern
of motion which is repeated every period of the external drive. Likewise, we can represent
period-2 motion as , where and represent distinguishable patterns of motion
which are repeated every alternate period of the external drive. A period-doubling
bifurcation is represented:
. Clearly, all that happens
in such a bifurcation is that the pendulum suddenly decides to do something slightly different in
alternate periods of the external drive.

Figure 40 shows the time-asymptotic phase-space orbit of the pendulum calculated for a
value of somewhat higher than that required to trigger the above mentioned period-doubling bifurcation. It can
be seen that the orbit is left-favouring (i.e., it spends the majority of its time on
the left-hand side of the plot), and takes the form of a closed curve consisting of
two interlocked loops in phase-space.
Recall that for period-1 orbits there was only a single closed loop in phase-space.
Figure 41 shows the Poincaré section of the orbit shown in Fig. 40.
The fact that the section consists
of two points confirms that the orbit does indeed correspond to period-2 motion.

A period-doubling bifurcation is an example of temporal symmetry breaking. The equations of
motion of the pendulum are invariant under the transformation
, where
is the period of the external drive. In the low amplitude (i.e., linear) limit
the time-asymptotic motion of the pendulum always respects this symmetry. However, as we have just seen, in the
non-linear regime it is possible to obtain solutions which spontaneously break this symmetry.
Obviously, motion which repeats itself every two periods of the external drive is not
as temporally symmetric as motion which repeats every period of the drive.

Figure 42:The -coordinate of the Poincaré section of a time-asymptotic orbit
plotted against the quality-factor . Data
calculated numerically for
, ,
, and . Data is shown for two sets of initial
conditions: and (lower branch); and and (upper branch).

Figure 42 is essentially a continuation of Fig 34. Data is shown for two
sets of initial conditions which lead to motions converging on left-favouring (lower branch) and
right-favouring (upper branch) periodic attractors. We have already seen that the left-favouring
periodic attractor undergoes a period-doubling bifurcation at . It is clear from Fig. 42
that the right-favouring attractor undergoes a similar bifurcation at almost exactly the
same -value. This is hardly surprising since, as has already
been mentioned, for fixed physics parameters (i.e., , , ), the
left- and right-favouring attractors are mirror-images of one another.